Turán-Ramsey theorems and simple asymptotically extremal structures

Abstract

This paper is a continuation of [10], where P. Erdős, A. Hajnal, V. T. Sós, and E. Szemerédi investigated the following problem:

Assume that a so called forbidden graphL and a functionf(n)=o(n) are fixed. What is the maximum number of edges a graphG n onn vertices can have without containingL as a subgraph, and also without having more thanf(n) independent vertices?

This problem is motivated by the classical Turán and Ramsey theorems, and also by some applications of the Turán theorem to geometry, analysis (in particular, potential theory) [27–29], [11–13].

In this paper we are primarily interested in the following problem. Let (G n ) be a graph sequence whereG n hasn vertices and the edges ofG n are coloured by the colours χ1,...,χ r so that the subgraph of colour χυ contains no complete subgraphK pv , (v=1,...r). Further, assume that the size of any independent set inG n iso(n) (asn→∞). What is the maximum number of edges inG n under these conditions?

One of the main results of this paper is the description of a procedure yielding relatively simple sequences of asymptotically extremal graphs for the problem. In a continuation of this paper we shall investigate the problem where instead of α(G n )=o(n) we assume the stronger condition that the maximum size of aK p -free induced subgraph ofG n iso(n).

This is a preview of subscription content, access via your institution.

References

  1. [1]

    B. Bollobás:Extremal graph theory, Academic Press, London, 1978.

    Google Scholar 

  2. [2]

    B. Bollobás andP. Erdős: On a Ramsey-Turán type problem,Journal of Combinatorial Theory, B21 (1976), 166–168.

    Google Scholar 

  3. [3]

    W. G. Brown, P. Erdős andM. Simonovits: Extremal problems for directed graphs,Journal of Combinatorial Theory,15B(1) (1973), 77–93.

    Google Scholar 

  4. [4]

    W. G. Brown, P. Erdős andM. Simonovits: Multigraph extremal problems, in:Problemes Combinatoires et Théorie des Graphes (ed. J. Bermond et al.), CNRS Paris, 1978.

    Google Scholar 

  5. [5]

    W. G. Brown, P. Erdős, andM. Simonovits: Inverse extremal digraph problems,Finite and Infinite Sets, Colloq. Math. Soc. J. Bolyai37 Eger 1981, Akad. Kiadó, Budapest (1985), 119–156.

    Google Scholar 

  6. [6]

    W. G. Brown, P. Erdős, andM. Simonovits: Algorithmic Solution of Extremal Digraph Problems,Trans. Amer. Math. Soc. 292 (1985), 421–449.

    Google Scholar 

  7. [7]

    S. Burr, P. Erdős andL. Lovász: On graphs of Ramsey type,Ars Combinatoria,1 (1976), 167–190.

    Google Scholar 

  8. [8]

    P. Erdős: On some new inequalities concerning extremal properties of graphs,Theory of Graphs, Proc. Coll. Tihany, Hungary (ed. P. Erdős and G. Katona), Acad. Press., N. Y. 1968, 77–81.

    Google Scholar 

  9. [9]

    P. Erdős Remarks on a theorem of Ramsey,Bull. Res. Conne. Israel 7 (1957), 21–24.

    Google Scholar 

  10. [10]

    P. Erdős, A. Hajnal, Vera T.Sós andE. Szemerédi More results on Ramsey-Turán type problem Ramsey-Turán type problems,Combinatorica 3 (1983), 69–82.

    Google Scholar 

  11. [11]

    P. Erdős, A. Meir, Vera T. Sós andP. Turán: On some applications of graph theory I,Discrete Math. 2 (1972), (3) 207–228.

    Google Scholar 

  12. [12]

    P. Erdős, A. Meir, Vera T. Sós andP. Turán: On some applications of graph theory II.Studies in Pure Mathematics (presented to R. Rado) Academic Press, London, 1971, 89–99.

    Google Scholar 

  13. [13]

    P. Erdős, A. Meir, Vera T. Sós andP. Turán: On some applications of graph theory III,Canadian Math. Bulletin 15 (1972), 27–32.

    Google Scholar 

  14. [14]

    P. Erdős andC. A. Rogers: The construction of certain graphs,Canadian Journal of Math 1962.; or Art of Counting MIT PRESS.

  15. [15]

    P. Erdős andVera T. Sós: Some remarks on Ramsey's and Turán's theorems, in:Combin. Theory and Appl. (P. Erdős et al eds) Colloq. Math. Soc. J. Bolyai4 (1969), 395–404.

  16. [16]

    P. Erdős andA. H. Stone: On the structure of linear graphs,Bull. Amer. Math. Soc. 52 (1946), 1089–1091.

    Google Scholar 

  17. [17]

    P. Frankl andV. Rödl: Some Ramsey-Turán type results for hypergraphs,Combinatorica 8 (1988), 323–332.

    Google Scholar 

  18. [18]

    Motzkin, E. G. Straus: Maxima for graphs and a new proof of a theorem of Turán,Canadian Journal of Math. 17 (1965), 533–540.

    Google Scholar 

  19. [19]

    F. P. Ramsey On a problem of formal logic,Proc. London Math. Soc. 2nd Series,30 (1930), 264–286.

    Google Scholar 

  20. [20]

    M. Simonovits: A method for solving extremal problems in graph theory, in:Theory of graphs, Proc. Coll. Tihany, (1966), (Ed. P. Erdős and G. Katona) Acad. Press, N.Y., 1968, 279–319.

    Google Scholar 

  21. [21]

    M. Simonovits: Extremal Graph Theory, in:Selected Topics in Graph Theory (eds Beineke and Wilson) Academic Press, London, New York, San Francisco, 161–200. (1983).

    Google Scholar 

  22. [22]

    Vera T. Sós: On extremal problems in graph theory, in:Proc. Calgary International Conf. on Combinatorial Structures and their Application, (1969), 407–410.

  23. [23]

    E. Szemerédi: On graphs containing no complete subgraphs with 4 vertices (in Hungarian),Mat. Lapok 23 (1972), 111–116.

    Google Scholar 

  24. [24]

    E. Szemerédi: On regular partitions of graphs, in:Problemes Combinatoires et Théorie des Graphes (ed. J. Bermond et al.), CNRS Paris, 1978, 399–401.

    Google Scholar 

  25. [25]

    P. Turán On an extremal problem in graph theory,Matematikai Lapok 48 (1941), 436–452 (in Hungarian), (see also [30]).

    Google Scholar 

  26. [26]

    P. Turán: On the theory of graphs,Colloq. Math. 3 (1954), 19–30, (see also [30]).

    Google Scholar 

  27. [27]

    P. Turán: Applications of graph theory to geometry and potential theory, in:Proc. Calgary International Conf. on Combinatorial Structures and their Application, (1969), 423–434, (see also [30]).

  28. [28]

    P. Turán: Constructive theory of functions, in:Proc. Internat. Conference in Varna, Bulgaria, 1970 Izdat. Bolgar Akad. Nauk, Sofia, 1972, (see also [30]).

    Google Scholar 

  29. [29]

    P. Turán: A general inequality of potential theory,Proc. Naval Research Laboratory, Washington, (1970), 137–141, (see also [30]).

  30. [30]

    Collected works of Paul Turán, Akadémiai Kiadó, Budapest, 1989.

  31. [31]

    A. A. Zykov: On some properties of linear complexes, Mat Sbornik, 24 (1949), 163–188; Amer. Math. Soc. Translations79, 1952.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Erdős, P., Hajnal, A., Simonovits, M. et al. Turán-Ramsey theorems and simple asymptotically extremal structures. Combinatorica 13, 31–56 (1993). https://doi.org/10.1007/BF01202788

Download citation

AMS subject classification code (1991)

  • 05 C 35
  • 05 C 55